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On invariant subalgebras when the ISR property fails

Yongle Jiang, Ruoyu Liu

TL;DR

This work classifies all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$, showing that, in the absence of the ISR property, every invariant subalgebra is either of the form $L(H)$ for a normal subgroup $H\lhd G$ or an $A_n$-type algebra defined by a trace symmetry condition. The authors combine the deformation/rigidity framework (cds) with invariant subalgebra techniques (aho) to reduce to amenable cases and perform a detailed conditional expectation analysis on the subgroup generated by $s=-I_2$, obtaining a finite list of possibilities and a dichotomy. As a corollary, they identify $L(\mathbb{Z}^2\rtimes \{\pm I_2\})$ as the unique maximal Haagerup $G$-invariant subalgebra, illustrating maximal Haagerup phenomena in this non-ISR icc group. The results advance understanding of invariant von Neumann subalgebras beyond ISR groups and suggest avenues for applying the methodology to other bi-exact groups and wreath products.

Abstract

We classify all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$. This is the first result on classifying $G$-invariant von Neumann subalgebras in $L(G)$ for i.c.c. groups $G$ without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that $L(\mathbb{Z}^2\rtimes \{\pm I_2\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\mathbb{Z})$.

On invariant subalgebras when the ISR property fails

TL;DR

This work classifies all -invariant von Neumann subalgebras in for , showing that, in the absence of the ISR property, every invariant subalgebra is either of the form for a normal subgroup or an -type algebra defined by a trace symmetry condition. The authors combine the deformation/rigidity framework (cds) with invariant subalgebra techniques (aho) to reduce to amenable cases and perform a detailed conditional expectation analysis on the subgroup generated by , obtaining a finite list of possibilities and a dichotomy. As a corollary, they identify as the unique maximal Haagerup -invariant subalgebra, illustrating maximal Haagerup phenomena in this non-ISR icc group. The results advance understanding of invariant von Neumann subalgebras beyond ISR groups and suggest avenues for applying the methodology to other bi-exact groups and wreath products.

Abstract

We classify all -invariant von Neumann subalgebras in for . This is the first result on classifying -invariant von Neumann subalgebras in for i.c.c. groups without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that is the unique maximal Haagerup -invariant von Neumann subalgebra in , where denotes the identity matrix in .
Paper Structure (5 sections, 5 theorems, 22 equations)

This paper contains 5 sections, 5 theorems, 22 equations.

Table of Contents

  1. introduction
  2. Acknowledgements
  3. Preliminaries
  4. Proof of Theorem \ref{['thm']} and Corollary \ref{['cor']}
  5. Remarks and open questions

Key Result

Theorem 1.1

Let $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$. Then a von Neumann subalgebra $P\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\subseteq G$ or $P=A_n$ for some $n\geq 0$, where $A_n:=\{x\in L(n\mathbb{Z}^2):~\tau(xu_g)=\tau(xu_{g^{-1}}), ~\forall~g\in G\}$,

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • proof : Proof of the Claim
  • proof : Proof of the Observation
  • ...and 1 more